Does informing the mother and gaining her approval to take money from her salary, then giving it to the father, absolve the daughter of responsibility for the additional amount she used to ask from her father, or must she donate the amount after employment?

We understand from the question that you were asking your father for the teacher's fee, an amount greater than what the teacher deserved, deceiving him, and saving the excess for your father's needs. Therefore, you must return what you took in excess and did not give back to your father, or ask for his forgiveness. You can pay the amount from your mother's salary to your father in lieu of what you took from him unjustly, to clear your conscience. You are not required to inform him of this. It is sufficient for you to return the right to him and seek forgiveness.